Olympiad problems

2011 Hanoi Open Mathematics Competitions, 4

Tags: algebra , inequalities

Among the five statements on real numbers below, how many of them are correct? "If $a < b < 0$ then $a < b^2$" , "If $0 < a < b$ then $a < b^2$", "If $a^3 < b^3$ then $a < b$", "If $a^2 < b^2$ then $a < b$", "If $|a| < |b|$ then $a < b$", (A) $0$, (B) $1$, (C) $2$, (D) $3$, (E) $4$

2010 IMC, 1

Tags: integration , real analysis

Let $0 < a < b$. Prove that $\int_a^b (x^2+1)e^{-x^2} dx \geq e^{-a^2} - e^{-b^2}$.

2009 German National Olympiad, 2

Tags: number theory

Find all positive interger $ n$ so that $ n^3\minus{}5n^2\plus{}9n\minus{}6$ is perfect square number.

2015 AMC 10, 16

Tags: vieta , algebra , polynomial , symmetry , rational root theorem , system of equations

If $y+4 = (x-2)^2, x+4 = (y-2)^2$, and $x \neq y$, what is the value of $x^2+y^2$? $ \textbf{(A) }10\qquad\textbf{(B) }15\qquad\textbf{(C) }20\qquad\textbf{(D) }25\qquad\textbf{(E) }\text{30} $

2013 NIMO Problems, 3

Tags: college , search

At Stanford in 1988, human calculator Shakuntala Devi was asked to compute $m = \sqrt[3]{61{,}629{,}875}$ and $n = \sqrt[7]{170{,}859{,}375}$. Given that $m$ and $n$ are both integers, compute $100m+n$. [i]Proposed by Evan Chen[/i]

2000 National Olympiad First Round, 25

Tags: geometry

The area of a convex quadrilateral $ABCD$ is $18$. If $|AB|+|BD|+|DC|=12$, then what is $|AC|$? $ \textbf{(A)}\ 9 \qquad\textbf{(B)}\ 6\sqrt 3 \qquad\textbf{(C)}\ 8 \qquad\textbf{(D)}\ 6 \qquad\textbf{(E)}\ 6\sqrt 2 $

2019 Ukraine Team Selection Test, 3

Tags: geometry , circumcircle , orthocenter , altitude , tangent circle

Given an acute triangle $ABC$ . It's altitudes $AA_1 , BB_1$ and $CC_1$ intersect at a point $H$ , the orthocenter of $\vartriangle ABC$. Let the lines $B_1C_1$ and $AA_1$ intersect at a point $K$, point $M$ be the midpoint of the segment $AH$. Prove that the circumscribed circle of $\vartriangle MKB_1$ touches the circumscribed circle of $\vartriangle ABC$ if and only if $BA1 = 3A1C$. (Bondarenko Mykhailo)

2021 Science ON all problems, 4

Tags: equality case , set , combinatorics

Take $k\in \mathbb{Z}_{\ge 1}$ and the sets $A_1,A_2,\dots, A_k$ consisting of $x_1,x_2,\dots ,x_k$ positive integers, respectively. For any two sets $A$ and $B$, define $A+B=\{a+b~|~a\in A,~b\in B\}$. Find the least and greatest number of elements the set $A_1+A_2+\dots +A_k$ may have. [i] (Andrei Bâra)[/i]

OIFMAT II 2012, 2

Tags: number theory , functional , functional equation in n

Find all functions $ f: N \rightarrow N $ such that: $\bullet$ $ f (m) = 1 \iff m = 1 $; $\bullet$ If $ d = \gcd (m, n) $, then $ f (mn) = \frac {f (m) f (n)} {f (d)} $; and $\bullet$ $ \forall m \in N $, we have $ f ^ {2012} (m) = m $. Clarification: $f^n (a) = f (f^{n-1} (a))$

2011 ELMO Shortlist, 4

Tags: modular arithmetic , ellipse , geometric transformation , reflection , geometry , conic

Prove that for any convex pentagon $A_1A_2A_3A_4A_5$, there exists a unique pair of points $\{P,Q\}$ (possibly with $P=Q$) such that $\measuredangle{PA_i A_{i-1}} = \measuredangle{A_{i+1}A_iQ}$ for $1\le i\le 5$, where indices are taken $\pmod5$ and angles are directed $\pmod\pi$. [i]Calvin Deng.[/i]

2016 China Northern MO, 7

Tags: number theory

Define sequence $(a_n):a_n=2^n+3^n+6^n+1(n\in\mathbb{Z}_+)$. Are there intenger $k\geq2$, satisfying that $\gcd(k,a_i)=1$ for all $k\in\mathbb{Z}_+$? If yes, find the smallest $k$. If not, prove this.

2005 Morocco National Olympiad, 2

Tags: quadratic , induction , number theory

Find all the positive integers $x,y,z$ satisfiing : $x^{2}+y^{2}+z^{2}=2xyz$

2020 Thailand Mathematical Olympiad, 4

Tags: geometry

Let $\triangle ABC$ be a triangle with altitudes $AD,BE,CF$. Let the lines $AD$ and $EF$ meet at $P$, let the tangent to the circumcircle of $\triangle ADC$ at $D$ meet the line $AB$ at $X$, and let the tangent to the circumcircle of $\triangle ADB$ at $D$ meet the line $AC$ at $Y$. Prove that the line $XY$ passes through the midpoint of $DP$.

2014 Stars Of Mathematics, 1

Tags: number theory , diophantine equation

Prove that for any integer $n>1$ there exist infinitely many pairs $(x,y)$ of integers $1<x<y$, such that $x^n+y \mid x+y^n$. ([i]Dan Schwarz[/i])

2019 Ramnicean Hope, 2

Tags: complex number , algebra

Let be three complex numbers $ a,b,c $ such that $ |a|=|b|=|c|=1=a^2+b^2+c^2. $ Calculate $ \left| a^{2019} +b^{2019} +c^{2019} \right| . $ [i]Costică Ambrinoc[/i]

2016 India Regional Mathematical Olympiad, 2

Tags: maximum , algebra , combinatorics

On a stormy night ten guests came to dinner party and left their shoes outside the room in order to keep the carpet clean. After the dinner there was a blackout, and the gusts leaving one by one, put on at random, any pair of shoes big enough for their feet. (Each pair of shoes stays together). Any guest who could not find a pair big enough spent the night there. What is the largest number of guests who might have had to spend the night there?

2020 Kazakhstan National Olympiad, 2

Tags: algebra , functional equation

Find all functions $ f: \mathbb {R} ^ + \to \mathbb {R} ^ + $ such that for any $ x, y \in \mathbb {R} ^ + $ the following equality holds: \[f (x) f (y) = f \left (\frac {xy} {x f (x) + y} \right). \] $ \mathbb {R} ^ + $ denotes the set of positive real numbers.

2012-2013 SDML (Middle School), 15

Tags: geometry

Pentagon $ABCDE$ is inscribed in a circle such that $ACDE$ is a square with area $12$. What is the largest possible area of pentagon $ABCDE$? $\text{(A) }9+3\sqrt{2}\qquad\text{(B) }13\qquad\text{(C) }12+\sqrt{2}\qquad\text{(D) }14\qquad\text{(E) }12+\sqrt{6}-\sqrt{3}$

2024 USAMO, 3

Tags: geometry

Let $m$ be a positive integer. A triangulation of a polygon is [i]$m$-balanced[/i] if its triangles can be colored with $m$ colors in such a way that the sum of the areas of all triangles of the same color is the same for each of the $m$ colors. Find all positive integers $n$ for which there exists an $m$-balanced triangulation of a regular $n$-gon. [i]Note[/i]: A triangulation of a convex polygon $\mathcal{P}$ with $n \ge 3$ sides is any partitioning of $\mathcal{P}$ into $n-2$ triangles by $n-3$ diagonals of $\mathcal{P}$ that do not intersect in the polygon's interior. [i]Proposed by Krit Boonsiriseth[/i]

2010 Contests, 2

Tags: trigonometry , limit , real analysis

Calculate the sum of the series $\sum_{-\infty}^{\infty}\frac{\sin^33^k}{3^k}$.

2007 Estonia Team Selection Test, 5

Tags: functional , continuous , functional equation , algebra

Find all continuous functions $f: R \to R$ such that for all reals $x$ and $y$, $f(x+f(y)) = y+f(x+1)$.

2010 Turkey MO (2nd round), 1

Tags: ceiling function , combinatorics

In a country, there are some two-way roads between the cities. There are $2010$ roads connected to the capital city. For all cities different from the capital city, there are less than $2010$ roads connected to that city. For two cities, if there are the same number of roads connected to these cities, then this number is even. $k$ roads connected to the capital city will be deleted. It is wanted that whatever the road network is, if we can reach from one city to another at the beginning, then we can reach after the deleting process also. Find the maximum value of $k.$

2017 Azerbaijan Junior National Olympiad, P1

Tags: system of equations , algebra

Solve the system of equation for $(x,y) \in \mathbb{R}$ $$\left\{\begin{matrix} \sqrt{x^2+y^2}+\sqrt{(x-4)^2+(y-3)^2}=5\\ 3x^2+4xy=24 \end{matrix}\right.$$ \\ Explain your answer

India EGMO 2025 TST, 2

Tags: algebra , polynomial

Two positive integers are called anagrams if every decimal digit occurs the same number of times in each of them (not counting the leading zeroes). Find all non-constant polynomials $P$ with non-negative integer coefficients so that whenever $a$ and $b$ are anagrams, $P(a)$ and $P(b)$ are anagrams as well. Proposed by Sutanay Bhattacharya

2008 Dutch IMO TST, 5

Tags: geometry , semicircle , midpoint

Let $\vartriangle ABC$ be a right triangle with $\angle B = 90^o$ and $|AB| > |BC|$, and let $\Gamma$ be the semicircle with diameter $AB$ that lies on the same side as $C$. Let $P$ be a point on $\Gamma$ such that $|BP| = |BC|$ and let $Q$ be on $AB$ such that $|AP| = |AQ|$. Prove that the midpoint of $CQ$ lies on $\Gamma$.